Bulletin of the Malaysian Mathematical Sciences Society ( IF 1.0 ) Pub Date : 2020-07-21 , DOI: 10.1007/s40840-020-00974-z Jin-Cai Kang , Yong-Yong Li , Chun-Lei Tang
In this paper, we study the following Chern–Simons–Schrödinger equation
$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\Delta u+\omega u+\lambda \Big (\frac{h^{2}(|x|)}{|x|^{2}}+ \int _{|x|}^{+\infty }\frac{h(s)}{s}u^{2}(s)\hbox {d}s\Big )u=g(u) \quad \text{ in }\ {\mathbb {R}}^{2},\\ \displaystyle u\in H_r^1({\mathbb {R}}^{2}), \end{array}\right. } \end{aligned}$$where \(\omega ,\lambda >0\) and \(h(s)=\frac{1}{2}\int _{0}^{s}ru^{2}(r)\hbox {d}r\). Since the nonlinearity g is asymptotically 5-linear at infinity, there would be a competition between g and the nonlocal term. By constrained minimization arguments and the quantitative deformation lemma, we prove the existence of least energy sign-changing radial solution, which changes sign exactly once. Further, we study the concentration of the least energy sign-changing radial solutions as \(\lambda \rightarrow 0\).
中文翻译：
具有渐近5线性非线性的Chern–Simons–Schrödinger方程的符号改变解
在本文中，我们研究以下Chern–Simons–Schrödinger方程
$$ \ begin {aligned} {\ left \ {\ begin {array} {ll} \ displaystyle-\ Delta u + \ omega u + \ lambda \ Big（\ frac {h ^ {2}（| x |）} {| x | ^ {2}} + \ int _ {| x |} ^ {+ \ infty} \ frac {h（s）} {s} u ^ {2}（s）\ hbox {d} s \ Big） u = g（u）\ quad \ text {in} \ {\ mathbb {R}} ^ {2}，\\ \ displaystyle u \ in H_r ^ 1（{\ mathbb {R}} ^ {2}）， \ end {array} \ right。} \ end {aligned} $$其中\（\ omega，\ lambda> 0 \）和\（h（s）= \ frac {1} {2} \ int _ {0} ^ {s} ru ^ {2}（r）\ hbox {d } r \）。由于非线性g在无穷大处渐近为5线性，因此g与非局部项之间将存在竞争。通过约束极小化论证和定量变形引理，我们证明了最小能量符号改变径向解的存在，该能量解正好改变一次符号。此外，我们研究了能量变化最小的径向解的浓度为\（\ lambda \ rightarrow 0 \）。
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